The Critical Exponent of the Helicity Modulus of the ${\cal O}(3)$ classical Heisenberg Model

Classical 3 dimensional spins in a 3 dimensional lattice interacting with a ferromagnetic interaction are a well-known model for a ferromagnetic material. They also are interesting as a field-theoretic model, where the lattice points can be considered points in space and the spin directions the coordinates for a propagating field.

When the spins in some volume are in thermodynamic equilibrium at any given temperature, one can imagine ``twisting'' the spins of one boundary surface through some angle (while holding them fixed) and then allowing the interior spins to achieve a new thermal equilibrium across the twist. What one has done is artificially create a fluctuation away from the equilibrium with a specific average helicity.

At high temperatures (above the Curie critical temperature) such a fluctuation costs no free energy - the system is fluid and there is no sheer modulus to restore a twist across the volume. At low temperatures the system is ``stiff'' - a helicity wave (or spin wave) costs a specific amount in free energy, corresponding to a nonzero shear or helicity modulus.

At the critical temperature the helicity modulus is quite interesting. This is the temperature where the spin system becomes in some sense ``stiff'' - long range spin correlations cause the free energy to increase even though the boundary spins being twisted are separated by a long distance. The helicity modulus itself has a critical exponent $x$ associated with its discontinuous appearance at $T_c$.

This exponent can be related to other exponents by scaling and renormalization arguments, and some of the other exponents (especially $\alpha$, the specific heat exponent) are remarkably difficult to numerically measure with e.g. Monte Carlo methods at high precision for the ${\cal O}(3)$ model (because the exponents are very small - so small that it is difficult to resolve with absolute assurance even the sign of the exponent let alone its magnitude. It thus appears that a high precision measurement of the helicity exponent $x$ would provide a valuable indirect measurement of $\alpha$, while having some considerable interest in its own right.

We are currently evaluating the helicity exponent with direct Monte Carlo and finite size scaling methods. These involve some very time and resource consuming (beowulf-based) computations that will undoubtedly continue for months to years as we seek sufficient precision to be able to unambiguously answer the questions associated with the measurement. Our best results to date (still unpublished) are presented in figure 1 below, where we are currently working on a point for $L = 64$.

Figure 1: Finite size scaling plot for Helicity Modulus Exponent $x$.